Smoothing of real algebraic hypersurfaces by rigid isotopies

Nabutovsky, Alexander

Annales de l'Institut Fourier, Tome 41 (1991) no. 1, pp. 11-25.

Soit $M_n\subset {\mathbf{R}}^{n+1}$ une hypersurface compacte lisse. Nous définissons $\kappa \left(M^n\right)$ comme le rapport ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$ où $r\left(M^n\right)$ est la distance de $M^n$ à l’ensemble central de $M^n$ (en d’autres termes, $r\left(M^n\right)$ est le rayon maximal d’un voisinage tubulaire ouvert de $M^n$ sans self-intersection). Nous prouvons que chaque hypersurface algébrique réelle non-singulière de degré $d$ peut être liée par une isotopie rigide avec une hypersurface algébrique ${\Sigma }_0^n$ de degré $d$ telle que $\kappa \left({\Sigma }_0^n\right)\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Ici $c\left(n\right)$, $\alpha \left(n\right)$ ne dépendent que de $n$, et isotopie rigide est une isotopie qui passe seulement à travers des hypersurfaces algébriques de degré $\le d$.

Comme application de ce résultat, nous démontrons qu’il existe des constantes $c,\beta$ telles que chaque paire de courbes planaires algébriques réelles non-singulières de degré $d$ peut être liée par une isotopie qui passe à travers des courbes algébriques de degré $\le exp\left(c{d}^{\beta d^2}\right)$. On en déduit par ailleurs, pour $n$ fixé, une borne supérieure en fonction de $d$, du nombre minimal de simplexes dans une triangulation $C^{\infty }$ d’une hypersurface algébrique de dimension $n$, non singulière de degré $d$.

Define for a smooth compact hypersurface $M^n$ of ${\mathbf{R}}^{n+1}$ its crumpleness $\kappa \left(M^n\right)$ as the ratio ${diam}_{{\mathbf{R}}^{n+1}}(M^n)/r(M^n)$, where $r\left(M^n\right)$ is the distance from $M^n$ to its central set. (In other words, $r\left(M^n\right)$ is the maximal radius of an open non-selfintersecting tube around $M^n$ in ${\mathbf{R}}^{n+1}.)$

We prove that any $n$-dimensional non-singular compact algebraic hypersurface of degree $d$ is rigidly isotopic to an algebraic hypersurface of degree $d$ and of crumpleness $\le exp\left(c\right(n\left)d^{\alpha \left(n\right)d^{n+1}}\right)$. Here $c\left(n\right)$, $\alpha \left(n\right)$ depend only on $n$, and rigid isotopy means an isotopy passing only through hypersurfaces of degree $\le d$. As an application, we show that for some constants $c,\beta$ any two isotopic smooth non-singular algebraic compact curves of degree $\le d$ in ${\mathbf{R}}^2$ can be connected by an isotopy passing only through algebraic curves of degree $\le exp\left(c{d}^{\beta d^2}\right)$. As another application, we show how to derive an upper bound in terms of $d$ only (for a fixed $n$) for the minimal number of simplices in a $C^{\infty }$- triangulation of a compact non-singular $n$-dimensional algebraic hypersurface of degree $d$.

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